Fermions and bosons, the only particles allowed in three dimensions, obey simple exchange rules. For example, if you swap the positions of two electrons the wavefunction is multiplied by -1. The multiplier is +1 for bosons. Since we need to square the wavefunction amplitude for any measurement, we cannot tell if we swapped two fermions or two bosons – they are indistinguishable.
In low dimensional systems (2D or 1D) other types of particles are allowed. Those for which the wavefunction is multiplied by a factor other than +1 or -1 are called ‘anyons’ because the multiplier can, in principle, be any complex number of absolute value 1. Such anyons, for example, are associated with the fractional quantum Hall effect. If the wavefunction is multiplied by a factor, the anyons are known as Abelian because the order in which multiple anyons are swapped does not matter, the final wavefunction is obtaine from the initial one by multiplying it with a product of factors from each swap.
We are interested in non-Abelian anyons. When such particles are swapped the wavefunction transforms and enters a different state that can be distinguished by measurement. The indistinguishability principle for fermions and bosons does not work. The most basic non-Abelian anyon is a Majorana Zero Mode, we are studying this anyon in semiconductor nanowires coupled to superconductors. There are many other non-Abelian particles such as parafermions (or fractional Majorana fermions), Fibonacci anyons etc. We are looking for ways to expand our research to discover and study these new particles. Non-abelian states have inspired the topological quantum computing architecture in which computation consists of moving anyons around each other (braiding).
Even supercomputers cannot fully compute the full wavefunction of tens of particles. This led to the idea of using well-controlled quantum states to simulate other unexplored quantum systems. For example, can a lattice of ultracold atoms be used to simulate a high temperature superconductor? Our approach to quantum simulation is to use arrays or chains of quantum dots in semiconductors. We control coupling between the dots and energy levels of each dot to construct a Hamiltonian that we would like to study. Of particular interest to us are Hamiltonians describing topological and non-Abelian states such as the Kitaev chain model.
Hybrid Quantum Bits
Quantum bits, or qubits, are the units of quantum information storage and processing. Many different realizations of a qubit were demonstrated in the past 20 years. However, the quest for the ultimate qubit continues because we are still interested in longer lived, faster and more scalable qubit designs. We are interseted in building qubits out of combinations of semiconductors and superconductors. Superconductors allow for the long-lived quantum states, while semiconductors offer fast control and scalability, as demonstrated by the CMOS technology. Combining the two can let us build conceptually new quantum bits suitable for topological quantum computing with non-Abelian states.